New Lower Bounds for the Number of Pseudoline Arrangements

09/10/2018 ∙ by Adrian Dumitrescu, et al. ∙ University of Wisconsin-Milwaukee 0

Arrangements of lines and pseudolines are fundamental objects in discrete and computational geometry. They also appear in other areas of computer science, such as the study of sorting networks. Let B_n be the number of arrangements of n pseudolines and let b_n=B_n. The problem of estimating B_n was posed by Knuth in 1992. Knuth conjectured that b_n ≤n 2 + o(n^2) and also derived the first upper and lower bounds: b_n ≤ 0.7924 (n^2 +n) and b_n ≥ n^2/6 -O(n). The upper bound underwent several improvements, b_n ≤ 0.6988 n^2 (Felsner, 1997), and b_n ≤ 0.6571 n^2 (Felsner and Valtr, 2011), for large n. Here we show that b_n ≥ cn^2 -O(n n) for some constant c>0.2083. In particular, b_n ≥ 0.2083 n^2 for large n. This improves the previous best lower bound, b_n ≥ 0.1887 n^2, due to Felsner and Valtr (2011). Our arguments are elementary and geometric in nature. Further, our constructions are likely to spur new developments and improved lower bounds for related problems, such as in topological graph drawings.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1 Introduction

Arrangement of Pseudolines.

A pseudoline in the Euclidean plane is a curve extending from negative infinity to positive infinity. An arrangement of pseudolines is a family of pseudolines where each pair of pseudolines has a unique point of intersection (called ‘vertex’). An arrangement is simple if no three pseudolines have a common point of intersection, see Fig. 1 (left). Here the term arrangement always means simple arrangement if not specified otherwise. Two arrangements are isomorphic if they can be mapped onto each other by a homeomorphism of the plane; see Fig. 1.

Figure 1: Left: A simple arrangement . Center: Wiring diagram of . Right: An arrangement that is not isomorphic to the arrangement on the left.

There are several representations and encodings of pseudoline arrangements. These representations help one count the number of arrangements. Three classic representations are: allowable sequences (introduced by Goodman and Pollack, see, e.g., [9, 10]), wiring diagrams (see for instance [7]), and zonotopal tilings (see for instance [6]). A wiring diagram is an Euclidean arrangement of pseudolines consisting of piece-wise linear ‘wires’, each horizontal except for a short segment where it crosses another wire. Each pair of wires cross only once. The wiring diagram in Fig. 1 (center) represents the arrangement . The above representations have been shown to be equivalent; bijective proofs to this effect can be found in [6]. Wiring diagrams are also known as reflection networks,i.e., networks the bring wires labeled from to into their reflection by means of performing switches of adjacent wires; see [13, p. 35]. Lastly, they are also known under the name of primitive sorting networks; see [14, Ch. 5.3.4]. The number of such networks is denoted by (a closed formula is given later in this section).

Let the number of nonisomorphic arrangements of pseudolines be denoted by . We are interested in the growth rate of ; so let . Knuth [13] conjectured that ; see also [7, p. 147] and [5, p. 259]. This conjecture is still open.

Upper bounds on the number of pseudoline arrangements.

Felsner [5] used a horizontal encoding of an arrangement in order to estimate . An arrangement can be represented by a sequence of horizontal cuts. The th cut is the list of the pseudolines crossing the th pseudoline in the order of the crossings. Using this approach, Felsner [5, Thm. 1] obtained the upper bound . The author [5, Thm. 2] refined this bound by using replace matrices. A replace matrix is a binary matrix with the properties for all and for all . Using this technique, he established the upper bound .

In his seminal paper on the topic, Knuth [13] took a vertical approach for encoding. Let be an arrangement of pseudolines. Adding a th pseudoline to , we get , an arrangement of pseudolines. The course of the th pseudoline describes a vertical cutpath from top to bottom. The number of arrangements of such that is isomorphic to is exactly the number of cutpaths in . Let denote the number of cutpaths in an arrangement of pseudolines. Therefore, one has ; and . Knuth [13] proved that , concluding that and thus ; this computation can be streamlined so that it yields , see [8]. Knuth also conjectured that , but this was refuted by Ondr̆ej Bílka in 2010 [8]; see also [7, p. 147]. Felsner and Valtr [8] proved a refined result, , by a careful analysis. This yields , which is the current best upper bound.

Lower bounds on the number of pseudoline arrangements.

Knuth [13, p. 37] gave a recursive construction in the setting of reflection networks. The number of nonisomorphic arrangements in his construction, thus also , obeys the recurrence

By induction this yields .

Matous̆ek sketched a simple—still recursive—grid construction in his book [17, Sec. 6.2], see Fig 2. Let be a multiple of and (assume that

is odd). The

lines in the two extreme bundles form a regular grid of points. The lines in the central bundle are incident to of these grid points. At each such point, there are choices; going below it or above it, thus creating at least binary choices. Thus obeys the recurrence

which by induction yields .

Figure 2: Grid construction for a lower bound on .

Felsner and Valtr [8] used rhombic tilings of a centrally symmetric hexagon in an elegant recursive construction for a lower bound on . Consider a set of pseudolines partitioned into the following three parts: , , , see Fig. 3111Figure reproduced from [8].. A partial arrangement is called consistent if any two pseudolines from two different parts always cross but any two pseudolines from the same part never cross.

Figure 3: The hexagon with one of its rhombic tilings and a consistent partial arrangement corresponding to the tiling.

The zonotopal duals of consistent partial arrangements are rhombic tilings of the centrally symmetric hexagon with side lengths . The enumeration of rhombic tilings of was solved by MacMahon [16] (see also [4]), who showed that the number of tilings is

(1)

A nontrivial (and quite involved) derivation using integral calculus shows that

Assuming to be a multiple of in the recursion step, the construction yields the lower bound recurrence

(2)

By induction, the analytic solution to formula (1) together with the recurrence (2) yield the lower bound , where In particular, for large ; this is the previous best lower bound.

Let denote the total number of wiring diagrams with wires (also known as reflection networks), and then is also the corresponding number of equivalence classes (see [13, p. 35]). Stanley [19] established the following closed formula for :

Table 1 shows the exact values of and , and their growth rate (up to four digits after the decimal point) with respect to , for small values of . The values of for to are from [13, p. 35] and the values of , , and are from [5][20] and [18], respectively; the values of , , and have been added recently, see [12, 18]. Observe that grows much faster than .

Table 1: Values of and for small .

Our results.

Here we extend the method used by Matoušek in his grid construction; observe that it uses lines of slopes. In Sections 2 (the 2nd part) and 3, we use lines of and different slopes in hexagonal type constructions; yielding lower bounds and for large , respectively. In Sections A and B, we use lines of and different slopes in rectangular type constructions; yielding the lower bounds and for large , respectively. While the construction in Section 3 gives a better bound, the one in Section 2 is easier to analyze. For each of the two styles, rectangular and hexagonal, the constructions are presented in increasing order of complexity. Our main result is summarized in the following.

Theorem 1.

Let be the number of arrangements of pseudolines. Then , for some constant . In particular, for large .

Outline of the proof.

We construct a line arrangement using lines of different slopes (for a small ). Let or (whichever is odd). Each bundle consists of equidistant lines in the corresponding parallel strip; remaining lines are discarded, or not used in the counting. A crossing point is an -wise crossing if it is incident to exactly lines. Let denote the number -wise crossings where each bundle consists of lines. Our goal is to estimate for each . Then we can locally replace the lines around each -wise crossing with any of the nonisomorphic pseudoline arrangements; and further apply recursively this construction to each of the bundles of parallel lines exiting this junction. This yields a simple pseudoline arrangement for each possible replacement choice. Consequently, the number of nonisomorphic pseudoline arrangements in this construction, say, , satisfies the recurrence:

(3)

where is a multiplicative factor counting the number of choices in this junction.

Related work.

In a comprehensive recent paper, Kync̆l [15] obtained estimates on the number of isomorphism classes of simple topological graphs that realize various graphs

. The author remarks that it is probably hard to obtain tight estimates on this quantity, “given that even for pseudoline arrangements, the best known lower and upper bounds on their number differ significantly”. While our improvements aren’t spectacular, it seems however likely that some of the techniques we used here can be employed to obtain sharper lower bounds for topological graph drawings too.

Notations and formulas used.

For two similar figures , let denote their similarity ratio. For a planar region , let denote its area. By slightly abusing notation, let denote the area of the triangle made by three lines , and . Assume that the equations of the three lines are , for , respectively. Then

Let denote the parallelogram made by the pairs of parallel lines and . Throughout this paper, is the logarithm in base of .

2 Preliminary constructions

Warm-up: a rectangular construction with slopes.

We start with a simple rectangular construction with bundles of parallel lines whose slopes are ; see Fig. 4. Let be the unit square we work with. The axes of all parallel strips are all incident to the center of . The final construction is obtained by a small clockwise rotation, so that there are no vertical lines.

Figure 4: Construction with slopes; here . The unit square is shown in blue.

Let , for , denote the area of the region covered by exactly of the strips. It is easy to see that , and obviously . Observe that is proportional to , for ; taking the boundary effect into account, we have

Now we can derive the multiplicative factor in Equation (3) as follows:

By induction on , the resulting lower bound is ; thereby this matches the constant, , in Knuth’s lower bound described in Section 1.

Hexagonal construction with slopes.

This construction yields the lower bound for large . Let be a regular hexagon of unit side. Consider bundles of parallel lines whose slopes are ; see Fig. 5 (left). Three parallel strips are bounded by the pairs of lines supporting opposite sides of , while the other three parallel strips are bounded by the pairs of lines supporting opposite short diagonals of . The axes of all six parallel strips are all incident to the center of the circle.

Assume a coordinate system where the lower left corner of is at the origin, and the lower side of lies along the -axis. Let be the partition of into six bundles of parallel lines. The lines in are contained in the parallel strip bounded by the two lines and , for . The equation of line is , with , , given in Fig 5 (right).

Figure 5: Left: The six parallel strips and corresponding covering multiplicities. These numbers only show incidences at the -wise crossings made by primary lines. Right: Coefficients of the lines .

We refer to lines in as the primary lines, and to lines in as secondary lines. The final construction is obtained by a small clockwise rotation, so that there are no vertical lines. Observe that

  • the distance between consecutive lines in any of the bundles of primary lines is ;

  • the distance between consecutive lines in any of the bundles of secondary lines is .

Let and denote the basic parallelogram (resp., triangle) determined by the lines in ; the side length of is . Let be the smaller regular hexagon made by the short diagonals of ; the similarity ratio is equal to . Recall that (i) the area of an equilateral triangle of side is ; and (ii) the area of a regular hexagon of side is ; as such, we have

Let , for , denote the area of the region covered by exactly of the strips. The following observations are in order: (i) the six isosceles triangles based on the sides of inside have unit base and height ; (ii) the six smaller equilateral triangles incident to the vertices of have side-length . These observations imply

Observe that . Recall that denote the number -wise crossings where each bundle consists of lines. Observe that is proportional to , for . Indeed, is equal to the number of -wise crossings of the primary lines that lie in a region covered by parallel strips, which is roughly equal to the ratio , for . More precisely, taking also the boundary effect of the relevant regions into account, we obtain

For estimating , the situation is little bit different, namely, in addition to considering -wise crossings of the primary lines, we also observe -wise crossings of the secondary lines at the centers of the small equilateral triangles contained in . See Fig. 6. It follows that

Figure 6: Triple incidences of secondary lines (drawn in red).

The values of , for , are summarized in Table 2; for convenience the linear terms are omitted. Since , can be also viewed as a function of .

Table 2: The values of and for .

Now we can derive the multiplicative factor in Equation (3) as follows:

We prove by induction on that for a suitable constant . It suffices to choose (using the values of for in Table 1) so that

The above inequality holds if we set , and the lower bound follows.

3 Hexagonal construction with slopes

This construction yields the lower bound for large , which is our main result in Theorem 1. Let be a regular hexagon of unit side. Consider bundles of parallel lines whose slopes are , , , , , . The axes of all parallel strips are all incident to the center of the circle created by the vertices of ; see Figs. 7 and 8.

Assume a coordinate system where the lower left corner of is at the origin, and the lower side of lies along the -axis. Let be the partition of into bundles of parallel lines. The lines in are contained in the parallel strip bounded by the two lines and , for . The equation of line is , with , , given in Table 3.

Table 3: Coefficients of the lines.

, and are bounded by the pairs of lines supporting opposite sides of , while , and are bounded by the pairs of lines supporting opposite short diagonals of . Therefore .

Figure 7: Construction with slopes. The twelve parallel strips and the corresponding covering multiplicities. These numbers only reflect incidences at the grid vertices made by the primary lines. The numbers inside are shown in Fig. 8
Figure 8: Construction with slopes. The twelve parallel strips and the corresponding covering multiplicities. These numbers only reflect incidences at the grid vertices made by the primary lines.

We refer to lines in as the primary lines, to lines in as the secondary lines, and to the rest of the lines in as the tertiary lines. The final construction is obtained by a small clockwise rotation, so that there are no vertical lines. Observe that the distance between consecutive lines in any of the bundles of

  • primary lines is ;

  • secondary lines is ;

  • tertiary lines is .

We refer to the intersection points of the primary lines as grid vertices. There are two types of grid vertices: the grid vertices in are intersection of primary lines and the ones outside are intersection of primary lines.

Let and denote the basic parallelogram (resp., triangle) determined by the primary lines (i.e., lines in ); the side length of is . We refer to these basic parallelograms as grid cell. Recall that (i) the area of an equilateral triangle of side is ; and (ii) the area of a regular hexagon of side is ; as such, we have

Let , for , denote the area of the region covered by exactly of the strips. Recall that denotes the area of the triangle made by , and .

Observe that is the area of the -gon . This -gon is not regular, since consecutive vertices lie on two concentric cycles of radii and centered at . So is the sum of the areas of congruent triangles; each with one vertex at the center of and other two as the two consecutive vertices of the -gon. Each of these triangles have area . Therefore,

Observe that the region whose area is consists of the hexagon and triangles outside . Therefore,

Recall that denotes the number -wise crossings where each bundle consists of lines. Observe that is proportional to , for ; indeed, is equal to the number of grid vertices that lie in a region covered by parallel strips, which is roughly equal to the ratio , for . More precisely, taking also the boundary effect of the relevant regions into account, we obtain

For , not all -wise crossings are at the grid vertices. There are types of such crossings in total; see Fig. 9. Types through are -wise crossings and types through are -wise crossings. The bundles intersecting at each of these types of vertices are listed in Table 4. For , let denote the weighted area containing all the crossings of type ; where the weight is the number of crossings per grid cell. To complete the estimates of for , we calculate for all from the bundles intersecting at crossings of type . The values are collected in Table 5. Observe that for two parallel strips and , the area of their intersection is ; recall that denotes the parallelogram made by the two pairs of parallel lines